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Pricing double barrier options under a volatility regime-switching model with psychological barriers

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Abstract

The prices of lots of assets have been proved in literature to exhibit special behaviors around psychological barriers, which is an important fact needed to be considered when pricing derivatives. In this paper, we discuss the valuation problem of double barrier options under a volatility regime-switching model where there exist psychological barriers in the prices of underlying assets. The volatility can shift between two regimes, that is to say, when the asset price rises up or falls down through the psychological barrier, the volatility takes two different values. Using the Laplace transform approach, we obtain the price of the double barrier knock-out call option as well as its delta. We also provide the eigenfunction expansion pricing formula and examine the effect of the psychological barrier on the option price and delta, finding that the gamma of the option is discontinuous at such barriers.

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Notes

  1. Readers should not be confused by the two kinds of barriers with different meanings in “double barrier option” and “psychological barrier”.

  2. We only find two previous papers that addressed discontinuous-coefficient diffusions for pricing purpose. One is Decamps et al. (2006) which used spectral expansion method to price interest rate derivatives in an economy subject to zero interest rate policy; the other is Gairat and Shcherbakov (2014) which provided the price formula for plain vanilla options grounded on the functionals of skew Brownian motions.

  3. The finding indicates that the investor should take extra care of his hedging strategy when the underlying asset breaks through the psychological barrier, which is in accordance with the perspective in Pierdzioch (2001): “Thus, combing the results of the theoretical and the empirical analysis, financial institutions or economic agents participating in the trading of options on one or the other of the mentioned currencies should take care of implicit price barriers when it comes either to the pricing or to the hedging of positions involving these contracts”. Such requirement for adjusting hedging positions is also advocated in Jang et al. (2015).

  4. We can suppose the price under the physical measure to be driven by

    $$\begin{aligned} \frac{{\mathrm {d}}S_t}{S_t}=\mu \left( S_t-S^{*}\right) {\mathrm {d}}t+\sigma \left( S_t-S^{*}\right) {\mathrm {d}}B_t, \end{aligned}$$

    where

    $$\begin{aligned} \mu (x) =\left\{ \begin{array}{ll} \displaystyle \mu _{1}> 0, &{} \quad x\ge 0,\\ \displaystyle \mu _{2}> 0, &{} \quad x< 0, \end{array} \right. \end{aligned}$$

    and the volatility function \(\sigma \) is of the form (2). The asymmetry in \(\mu \) consists with the market observation that the rates of return will change after the asset prices pass through psychological barriers as evidenced by Cyree et al. (1999) and Jang et al. (2015).

  5. This limitation of the model is also pointed out by the authors in their paper.

  6. Throughout this paper, we suppose that \(L \le S_{0}, K \le U\).

  7. The symmetric local time for a continuous semimartingale \(Y_{t}\) is defined by

    $$\begin{aligned} {L}^Y_{t}(x)=\lim _{\epsilon \rightarrow 0+}\frac{1}{2\epsilon }\int _{0}^{t}{\mathbbm {1}}_{\{|Y_s-x|\le \epsilon \}} {\mathrm {d}}\langle Y\rangle _{s}, \end{aligned}$$

    where \(\langle Y\rangle _{t}\) stands for the quadratic variation process of \(Y_{t}\). It is an increasing process and can be seen as an occupation time density. To see more details about this process, please refer to Philip (2004).

  8. The Mathematica code for this algorithm is available at http://www.pe.tamu.edu/valko/public_html/Nil/.

  9. Both of them can be implemented directly by using the built-in root-search function fzero and the numerical integration function quad in MATLAB 7.10, or FindRoot and NIntegrate in MATHEMATICA 8, respectively.

  10. From the definitions of \(g_{1}\) and \(g_{2}\) in (2), it is straightforward to know that \({\mathbf {E}}[e^{-\lambda T_{L,U}}]=g_{1}(X_{0})+g_{1}(X_{0})\) where \(X_{0}=\log (S_{0}/S^{*})\), which indicates that we can use numerical Laplace inversion to get the exit probability.

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Acknowledgements

The authors would like to thank the editors and the anonymous referee for their valuable comments and suggestions which helped to improve the paper significantly. This work was partially supported by the National Natural Science Foundation of China (No. 71532001).

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Authors and Affiliations

  1. School of Mathematics, Tianjin University, Tianjin, 300354, People’s Republic of China

    Shiyu Song

  2. School of Business, Nankai University, Tianjin, 300071, People’s Republic of China

    Yongjin Wang

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  1. Shiyu Song
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Correspondence to Shiyu Song.

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Song, S., Wang, Y. Pricing double barrier options under a volatility regime-switching model with psychological barriers. Rev Deriv Res 20, 255–280 (2017). https://doi.org/10.1007/s11147-017-9130-x

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